Software VNA and Microwave Network Design and Characterisation
Zhipeng Wu University of Manchester, UK
Software VNA and Microwave Network Design and Characterisation
Software VNA and Microwave Network Design and Characterisation
Zhipeng Wu University of Manchester, UK Copyright © 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone +44 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44) 1243 770620. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, ONT, Canada L5R 4J3 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Anniversary Logo Design: Richard J. Pacifico British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-470-51215-9 (HB) Typeset in 10/12pt Galliard by Integra Software Services Pvt. Ltd, Pondicherry, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. To Guoping, William and Richard
Contents
Foreword xv
Preface xvii
1 Introduction to Network Analysis of Microwave Circuits 1
1.1 One-Port Network 2 1.1.1 Total Voltage and Current Analyses 2 1.1.2 Transmission-Reflection Analysis 3 1.1.2.1 Voltage and current 3 1.1.2.2 Reflection coefficient 4 1.1.2.3 Power 5
1.1.2.4 Introduction of a1 and b1 5 1.1.2.5 Z in terms of 7 1.1.3 Smith Chart 7 1.1.3.1 Impedance chart 7 1.1.3.2 Admittance chart 8 1.1.4 Terminated Transmission Line 9 1.2 Two-Port Network 10 1.2.1 Total Quantity Network Parameters 10 1.2.2 Determination of Z, Y and ABCD Parameters 11 1.2.3 Properties of Z, Y and ABCD Parameters 12 1.2.4 Scattering Parameters 12 1.2.5 Determination of S-Parameters 14 1.2.6 Total Voltages and Currents in Terms of a and b Quantities 14 1.2.7 Power in Terms of a and b Quantities 14 1.2.8 Signal Flow Chart 15 viii CONTENTS
1.2.9 Properties of S-Parameters 15 1.2.10 Power Flow in a Terminated Two-Port Network 16 1.3 Conversions Between Z, Y and ABCD and S-Parameters 18 1.4 Single Impedance Two-Port Network 21 1.4.1 S-Parameters for Single Series Impedance 21 1.4.2 S-Parameters for Single Shunt Impedance 21 1.4.3 Two-Port Chart 22 1.4.3.1 Single series impedance network 22 1.4.3.2 Single shunt impedance network 23 1.4.3.3 Scaling property 25 1.4.4 Applications of Two-Port Chart 26 1.4.4.1 Identification of pure resonance 26 1.4.4.2 Q-factor measurements 27 1.4.4.3 Resonance with power-dependent losses 27 1.4.4.4 Impedance or admittance measurement using the two-port chart 28 1.5 S-Parameters of Common One- and Two-Port Networks 28 1.6 Connected Two-Port Networks 28 1.6.1 T-Junction 28 1.6.2 Cascaded Two-Port Networks 30 1.6.3 Two-Port Networks in Series and Parallel Connections 31 1.7 Scattering Matrix of Microwave Circuits Composed of One-Port and Multi-Port Devices 32 1.7.1 S-Parameters of a Multi-Port Device 32 1.7.2 S-Parameters of a Microwave Circuit 33 References 37
2 Introduction to Software VNA 39
2.1 How to Install 40 2.2 The Software VNA 41 2.3 Stimulus Functions 42 2.4 Parameter Functions 43 2.5 Format Functions 44 2.6 Response Functions 45 2.7 Menu Block 48 2.7.1 Cal 48 2.7.2 Display 48 2.7.3 Marker 51 2.7.4 DeltaM 53 2.7.5 Setting 54 2.7.6 Copy 55 CONTENTS ix
2.8 Summary of Unlabelled-Key Functions 55 2.9 Preset 56 2.10 Device Under Test 57 2.10.1 Device 59 2.10.2 Circuit 61 2.11 Circuit Simulator 63 2.11.1 Circuit Menu 63 2.11.2 Device Menu 64 2.11.3 Ports Menu 66 2.11.4 Connection Menu 67 2.12 Circuit Simulation Procedures and Example 67
3 Device Builders and Models 73
3.1 Lossless Transmission Line 74 3.2 One- and Two-Port Standards 76 3.3 Discrete RLC Components: One-Port Impedance Load 78 3.4 Discrete RLC Components: Two-Port Series Impedance 79 3.5 Discrete RLC Components: Two-Port Shunt Admittance 80 3.6 General Transmission Line 81 3.7 Transmission Line Components: Two-Port Serial Transmission Line Stub 82 3.8 Transmission Line Components: Two-Port Parallel Transmission Line Stub 83 3.9 Ideal Two-Port Components: Attenuator/Gain Block 85 3.10 Ideal Two-Port Components: 1:N and N:1 Transformer 86 3.11 Ideal Two-Port Components: Isolator 87 3.12 Ideal Two-Port Components: Gyrator 87 3.13 Ideal Two-Port Components: Circulator 88 3.14 Physical Transmission Lines: Coaxial Line 89 3.15 Physical Transmission Lines: Microstrip Line 90 3.16 Physical Transmission Lines: Stripline 94 3.17 Physical Transmission Lines: Coplanar Waveguide 96 3.18 Physical Transmission Lines: Coplanar Strips 98 3.19 Physical Line Discontinuities: Coaxial Line Discontinuities 101 3.19.1 Step Discontinuity 101 3.19.2 Gap Discontinuity 102 3.19.3 Open-End Discontinuity 103 3.20 Physical Line Discontinuities: Microstrip Line Discontinuities 104 3.20.1 Step Discontinuity 104 x CONTENTS
3.20.2 Gap Discontinuity 107 3.20.3 Bend Discontinuity 109 3.20.4 Slit Discontinuity 110 3.20.5 Open-End Discontinuity 110 3.21 Physical Line Discontinuities: Stripline Discontinuities 111 3.21.1 Step Discontinuity 111 3.21.2 Gap Discontinuity 114 3.21.3 Bend Discontinuity 115 3.21.4 Open-End Discontinuity 116 3.22 General Coupled Lines: Four-Port Coupled Lines 116 3.23 General Coupled Lines: Two-Port Coupled Lines 117 3.24 Physical Coupled Lines: Four-Port Coupled Microstrip Lines 119 3.25 Physical Coupled Lines: Two-Port Coupled Microstrip Lines 122 3.26 Lumped Elements: Inductors 123 3.26.1 Circular Coil 123 3.26.2 Circular Spiral 125 3.26.3 Single Turn Inductor 126 3.27 Lumped Elements: Capacitors 127 3.27.1 Thin Film Capacitor 127 3.27.2 Interdigital Capacitor 129 3.28 Lumped Elements: Resistor 129 3.29 Active Devices 130 3.30 Antennas: Dipole Antenna 130 3.31 Antennas: Resonant Antenna 134 3.32 Antennas: Transmission Between Dipole Antennas 135 3.33 Antennas: Transmission Between Resonant Antennas 136 3.34 User-Defined S-Parameters: One-Port Device 137 3.35 User-Defined S-Parameters: Two-Port Device 138 References 139
4 Design of Microwave Circuits 141
4.1 Impedance Matching 141 4.1.1 Impedance Matching Using a Discrete Element 141 4.1.2 Single Stub Matching 142 4.1.3 Double Stub Matching 143 4.2 Impedance Transformers 145 4.2.1 Quarter-Wave Transformer 145 4.2.2 Chebyshev Multisection Matching Transformer 146 4.2.3 Corporate Feeds 148 CONTENTS xi
4.3 Microwave Resonators 149 4.3.1 One-Port Directly Connected RLC Resonant Circuits 149 4.3.2 Two-Port Directly Connected RLC Resonant Circuits 150 4.3.3 One-Port Coupled Resonators 151 4.3.4 Two-Port Coupled Resonators 152 4.3.5 Transmission Line Resonators 154 4.3.6 Coupled Line Resonators 154 4.4 Power Dividers 155 4.4.1 The 3 dB Wilkinson Power Divider 155 4.4.2 The Wilkinson Power Divider with Unequal Splits 156 4.4.3 Alternative Design of Power Divider with Unequal Splits 157 4.4.4 Cohn’s Cascaded Power Divider 158 4.5 Couplers 159 4.5.1 Two-Stub Branch Line Coupler 159 4.5.2 Coupler with Flat Coupling Response 160 4.5.3 Three-Stub Branch Line Coupler 161 4.5.4 Coupled Line Couplers 162 4.6 Hybrid Rings 163 4.6.1 Hybrid Ring Coupler 163 4.6.2 Rat-Race Hybrid 164 4.6.3 Wideband Rat-Race Hybrid 164 4.6.4 Modified Hybrid Ring 165 4.6.5 Modified Hybrid Ring with Improved Bandwidth 165 4.7 Phase Shifters 166 4.7.1 Transmission Line Phase Shifter 166 4.7.2 LC Phase Shifters 167 4.8 Filters 168 4.8.1 Maximally Flat Response 168 4.8.2 Chebyshev Response 168 = 4.8.3 Maximally Flat Low-Pass Filters with c 1 169 = 4.8.4 Chebyshev Low-Pass Filters with c 1 171 4.8.5 Filter Transformations 172 4.8.6 Step Impedance Low-Pass Filters 173 4.8.7 Bandpass and Bandstop Filters Using /4 Resonators 174 4.8.8 Bandpass Filters Using /4 Connecting Lines and Short-Circuited Stubs 175 4.8.9 Coupled Line Bandpass Filters 176 4.8.10 End-Coupled Resonator Filters 178 xii CONTENTS
4.9 Amplifier Design 179 4.9.1 Maximum Gain Amplifier Design 179 4.9.2 Broadband Amplifier Design 181 4.9.3 High-Frequency Small Signal FET Circuit Model 182 4.9.4 Negative Feedback Amplifier Design 183 References 185
5 Simulation of Microwave Devices and Circuits 187
5.1 Transmission Lines 188 5.1.1 Terminated Transmission Line 188 5.1.2 Two-Port Transmission Line 189 5.1.3 Short-Circuited Transmission Line Stub 189 5.1.4 Open-Circuited Transmission Line Stub 190 5.1.5 Periodic Transmission Line Structures 192 5.2 Impedance Matching 194 5.2.1 Matching of a Half-Wavelength Dipole Antenna Using a Discrete Element 194 5.2.2 Single Stub Matching of a Half-Wavelength Dipole Antenna 195 5.3 Impedance Transformers 197 5.3.1 Quarter-Wave Impedance Transformer 197 5.3.2 Chebyshev Multisection Impedance Transformer 198 5.3.3 Corporate Feeds 199 5.3.4 Corporate Feeds Realised Using Microstrip Lines 201 5.3.5 Kuroda’s Identities 201 5.4 Resonators 205 5.4.1 One-Port RLC Series Resonant Circuit 205 5.4.2 Two-Port RLC Series Resonant Circuit 205 5.4.3 Two-Port Coupled Resonant Circuit 208 5.4.4 Two-Port Coupled Microstrip Line Resonator 208 5.4.5 Two-Port Coupled Microstrip Coupled Line Resonator 210 5.4.6 Two-Port Symmetrically Coupled Ring Resonator 212 5.4.7 Two-Port Asymmetrically Coupled Ring Resonator 213 5.5 Power Dividers 213 5.5.1 3 dB Wilkinson Power Divider 213 5.5.2 Microstrip 3 dB Wilkinson Power Divider 216 5.5.3 Cohn’s Cascaded 3 dB Power Divider 217 CONTENTS xiii
5.6 Couplers 219 5.6.1 Two-Stub Branch Line Coupler 219 5.6.2 Microstrip Two-Stub Branch Line Coupler 221 5.6.3 Three-Stub Branch Line Coupler 221 5.6.4 Coupled Line Coupler 222 5.6.5 Microstrip Coupled Line Coupler 225 5.6.6 Rat-Race Hybrid Ring Coupler 225 5.6.7 March’s Wideband Rat-Race Hybrid Ring Coupler 226 5.7 Filters 229 5.7.1 Maximally Flat Discrete Element Low-Pass Filter 229 5.7.2 Equal Ripple Discrete Element Low-Pass Filter 231 5.7.3 Equal Ripple Discrete Element Bandpass Filter 232 5.7.4 Step Impedance Low-Pass Filter 233 5.7.5 Bandpass Filter Using Quarter-Wave Resonators 236 5.7.6 Bandpass Filter Using Quarter-Wave Connecting Lines and Short-Circuited Stubs 236 5.7.7 Microstrip Coupled Line Filter 239 5.7.8 End-Coupled Microstrip Resonator Filter 241 5.8 Amplifier Design 241 5.8.1 Maximum Gain Amplifier 241 5.8.2 Balanced Amplifier 245 5.9 Wireless Transmission Systems 245 5.9.1 Transmission Between Two Dipoles with Matching Circuits 245 5.9.2 Transmission Between Two Dipoles with an Attenuator 249 References 249
Index 251
Foreword
Software VNA and Microwave Network Design and Characterisation is a unique contribution to the microwave literature. It fills a need in the education and training of microwave engineers and builds upon well-established texts such as Fields and Waves in Communication Electronics by S. Ramo, J. Whinnery and T. Van Duzer, Foundations for Microwave Engineering by R.E. Collin and Microwave Engineering by D. Pozar. The ‘virtual vector network analyser’ that can be downloaded from the CD supplied with the book enables those without access to a real instrument to learn how to use a vector network analyser. The many design examples provide opportunities for the reader to become familiar with the Software VNA and the various formats in which frequency responses can be displayed. They also encourage ‘virtual experiments’. Design formulas for many devices are given, but the underlying theory that can be found in other texts is not covered to avoid repetition. A circuit theory or field theory approach is available and this encourages the user to link the two. A novel feature of the book is the introduction and application of a two-port chart that complements the well-known Smith chart, widely used for one-port circuits. The two-port chart enables the frequency response of transmission parameters to be displayed as well as reflection parameters. The range of devices introduced in the book includes stubs, transformers, power dividers/combiners, couplers, filters, antennas and amplifiers. Nonideal behaviour, e.g. the effects of dielectric, conductor and radiation losses, is included for many devices. The devices can be connected to form microwave circuits and the frequency response of the circuit can be ‘measured’. The lower frequency limit in the Software VNA is 1 Hz and circuits containing both lumped and distributed devices can be characterised. Assuming a knowledge of transmission lines, circuits and some electromagnetic theory, Software VNA is suitable for introduction at the xvi FOREWORD final-year undergraduate level and postgraduate levels. Students would be stimulated by the opportunity to ‘measure’ their own devices and circuits. Experienced microwave engineers will also find Software VNA useful. L.E. Davis University of Manchester Preface
In addition to conventional textbooks, the advances in computer technology and modern microwave test instruments over the past decade have given electrical engineers, researchers and university students two new approaches to study microwave components, devices and circuits. The Vector Network Analyser (VNA) is one of the most desirable instruments in the area of microwave engineering, which can provide fast and accurate characterisation of microwave components, devices or circuits of interest. On the other hand, a commercial microwave circuit simulation software package offers a cost-effective way to study the properties of microwave components and devices before they are used to construct circuits and the properties of the circuits before they are built for testing. However, mainly due to their costs, VNAs and microwave circuit simulators are not widely accessible on a day-to-day basis to many electrical engineers, researchers and university students. This book together with the associated software is intended to fill in the gap between these two aspects with (i) an introduction to microwave network analysis, microwave components and devices, microwave circuit design and (ii) the provision of both device and circuit simulators powered by the analytical formulas published in the literature. The purpose of the associated software named Software VNA is fourfold. First, it functions as a VNA trainer with a lower frequency limit of 1 Hz and a upper frequency limit of 1000 GHz, providing to those who have not seen or used a VNA before the opportunity to have personal experience of how a VNA would operate in practice and be used for microwave measurements. Secondly, it provides experienced users with an option to get access to the data on a commercial VNA test instrument for data analysis, manipulation or comparison. Thirdly, it provides the users with a simulator equipped with 35 device builders from which an unlimited number of devices can be defined and studied. Analytical CAD equations, many of which have been experimentally verified, are used as models for simulation, giving no hidden xviii PREFACE numerical errors. The users may also use the Software VNA to verify the limitations and accuracy of the CAD equations. Finally, it provides the users with a circuit simulator that they can use to build circuits and study their properties. The book has five chapters. In Chapter 1, the basic theory of network analysis is introduced and network parameters are defined. In Chapter 2, the installation and functions of the Software VNA are described. In Chapter 3, the built-in device models are presented with detailed equations and their limitations. In Chapter 4, circuit design and operation principles for impedance matching, impedance transformation, resonators, power dividers, coupler, filters and amplifiers are summarised, and the design examples of these circuits are given in Chapter 5. The book and its associated software can be used for teaching in the area of microwave engineering. 1 Introduction to Network Analysis of Microwave Circuits
ABSTRACT
Network presentation has been used as a technique in the analysis of low- frequency electrical and electronic circuits. The same technique is equally useful in the analysis of microwave circuits, although different network parameters are used. In this chapter, network parameters for microwave circuit analysis, in particular scattering parameters, are introduced together with a Smith chart for one-port networks and a new chart for two-port networks. The analyses of two-port connected networks and a circuit composed of multi-port networks are also presented.
KEYWORDS
Network analysis, Network parameters, Impedance parameters, Admittance parameters, ABCD parameters, Scattering parameters, Smith chart, Two- port chart, Connected networks Network presentation has been used as a technique in the analysis of low-frequency electrical and electronic circuits (Ramo, Whinnery and van Duzer, 1984). The same technique is equally useful in the analysis of microwave circuits, although different network parameters may be used (Collin, 1966; Dobrowolski, 1991; Dobrowolski and Ostrowski, 1996; Fooks and Zakarev, 1991; Gupta, Garg and Chadha, 1981; Liao, 1990; Montgomery, Dicke and Purcell, 1948; Pozar, 1990; Rizzi, 1988; Ishii 1989; Wolff and Kaul, 1998). Using such a technique, a microwave circuit
Software VNA and Microwave Network Design and Characterisation Zhipeng Wu © 2007 John Wiley & Sons, Ltd 2 INTRODUCTION TO NETWORK ANALYSIS can be regarded as a network or a composition of a number of networks. Each network may also be composed of many elementary components. A network may have many ports, from which microwave energy flows into or out of the network. One- and two-port networks are, however, the most common, and most commercial network analysers provide measurements for one- or two-port networks. In this chapter, the network analysis will be based on one- and two-port networks. Network parameters, in particular scattering parameters, will be introduced together with a Smith chart for one-port networks and a new chart for two-port networks. The analysis of two connected networks and a circuit composed of a number of networks will also be presented. For further reading, see references at the end of the chapter.
1.1 ONE-PORT NETWORK
A one-port network can be simply represented by load impedance Z to the external circuit. When the network is connected to a sinusoidal voltage source with an open circuit peak voltage Vs and a reference internal impedance of Z0 ref as shown in Figure 1.1, the circuit can be analysed using the circuit theory based on total voltage and current quantities. It can also be analysed using the transmission-reflection analysis based on incident and reflected voltage and current quantities. Both analyses are described below.
The reference internal impedance Z0 ref of the source is assumed to be 50 throughout this book.
1.1.1 Total Voltage and Current Analyses
Using circuit theory, the voltage V on the load impedance and the current I flowing through it as shown in Figure 1.1 are related by
V = ZI 1.1
I
Z0,ref V Z Vs
Figure 1.1 Simplified one-port network ONE-PORT NETWORK 3 and they can be obtained by
= VsZ V + 1.2a Z0 ref Z and
= Vs I + 1.2b Z0 ref Z
The power delivered to the load impedance by the voltage source can be obtained by
1 V 2 P = Re VI∗ = s Re Z 1.3 L + 2 2 2 Z0 ref Z where ∗ indicates the complex conjugate.
1.1.2 Transmission-Reflection Analysis
1.1.2.1 Voltage and current
Using the transmission-reflection analysis, the incident voltage V + is defined to be the voltage that the voltage source could provide to a matched load, = + i.e. when Z Z0 ref , and the incident current I to be the corresponding current flowing through the matched load. Hence
V V + = s 1.4a 2 and
V + V I+ = = s 1.4b Z0 ref 2Z0 ref
= = + = + Therefore if ZL Z0 ref , then V V and I I . However, in the general = case that Z Z0 ref , the voltage V can be taken to be the superposition of two voltages: the incident voltage V + and a reflected voltage V −. Similarly the current I can be taken as the superposition of two currents: the incident + − = + current I and a reflected current I . Since V can be written as V IZ0 ref Ve with
V Z − Z = s 0 ref Ve + 1.5 Z Z0 ref 4 INTRODUCTION TO NETWORK ANALYSIS
– + I I
Z Z0,ref 0,ref
– + V V Vs Ve
Figure 1.2 Equivalent circuit the circuit in Figure 1.1 can be represented by an equivalent circuit shown in Figure 1.2. The load impedance Z is replaced by a ‘voltage source’ Ve with an ‘internal impedance’ Z0 ref . By using the superposition theorem, the reflected voltage can be taken to be that produced by the equivalent voltage source Ve so that Z − Z − = Ve = Vs 0 ref V + 1.6a 2 2 Z Z0 ref and − Z − Z − = V = Vs 0 ref I − 1.6b Z0 ref 2Z0 ref Z Z0 ref
The total voltage and current are then
Z = + + − = 0 ref V V V Vs + 1.7a Z Z0 ref and
= + − − = 1 I I I Vs + 1.7b Z Z0 ref which are the same as those in Equation (1.2) obtained using circuit theory.
1.1.2.2 Reflection coefficient
Using the transmission-reflection analysis, the total voltage is expressed as the sum of the incident voltage and the reflected voltage, and the current as the difference of the incident current and the reflected current. For the convenience of analysis, a reflection coefficient can be introduced to relate the reflected and incident quantities. The reflection coefficient is defined as
V − I− Z − Z Y − Y = = = 0 ref = 0 ref 1.8 + + + + V I Z Z0 ref Y0 ref Y ONE-PORT NETWORK 5
= = where Y0 ref 1/Z0 ref and Y 1/Z and is defined with respect to the reference impedance Z0 ref . The total voltage and current at the load can then be expressed as
V = V + 1 + 1.9a and
I = I+ 1 − 1.9b
Hence the total voltage and current quantities can be obtained when is determined.
1.1.2.3 Power
Associated with the incident voltage V + and the incident current I+ is the incident power which is given by
+2 2 + = 1 + +∗ = 1 V = Vs = P Re V I Pmax 1.10 2 2 Z0 ref 8Z0 ref
This power is also the maximum power available from the voltage source. Similarly the reflected power is associated with the reflected voltage V − and the reflected current I− and is given by
1 1 V −2 − = − −∗ = = 2 P Re V I Pmax 1.11 2 2 Z0 ref
The power delivered to the load impedance is the difference between the incident power and the reflected power, i.e.
= + − − = + − 2 PL P P P 1 1.12 which is identical to Equation (1.3).
1.1.2.4 Introduction of a1 and b1
+ Since the incident power is related to V and Z0 ref and the reflected power − to V and Z0 ref as in Equations (1.10) and (1.11), their expressions can be simplified with the introduction of two new quantities a1 and b1 which are defined as (Collin, 1966; Pozar, 1990)
+ = V a1 1.13a Z0 ref 6 INTRODUCTION TO NETWORK ANALYSIS and
− = V b1 1.13b Z0 ref
Using these two new quantities, the incident, reflected and total powers can then be expressed, respectively, as
1 P + = a 2 1.14a 2 1
1 P − = b 2 1.14b 2 1 and
1 P = a 2 −b 2 1.14c L 2 1 1
The voltage and current quantities can also be written as + = V a1 Z0 ref 1.15a
− = V b1 Z0 ref 1.15b
a I+ = 1 1.15c Z0 ref
b I− = 1 1.15d Z0 ref
= + V a1 b1 Z0 ref 1.15e and
a − b I = 1 1 1.15f Z0 ref ONE-PORT NETWORK 7
The reflection coefficient defined in Equation (1.8) becomes
Z − Z Y − Y = b1 = 0 ref = 0 ref + + 1.16 a1 Z Z0 ref Y0 ref Y
Using a1 and b1, the signal reflection property of the one-port network can be described by
= b1 a1 1.17
1.1.2.5 Z in terms of
The formulas derived above are useful for the analysis of one-port network when the load impedance is known. However, very often in practice, Z has to be determined from measurement. In this case, Z can be obtained from the measurement of the reflection coefficient using
1 + Z = Z 1.18 1 − 0 ref
1.1.3 Smith Chart
1.1.3.1 Impedance chart
The impedance Smith chart (Smith, 1939, 1944) is based on the expression of the reflection coefficient in terms of load impedance Z. With the introduction of the normalised load impedance with respect to the reference impedance Z0 ref as
Z z = = r + jx 1.19 Z0 ref where r and x are the normalised resistance and reactance, respectively, the reflection coefficient can be written as z − 1 r + jx − 1 = = = u + jv 1.20 z + 1 r + jx + 1 where u and v are the real and imaginary projections of on the complex u–v plane. Equation (1.20) can be rearranged to give the following two separate equations: r 2 1 u − + v2 = 1.21a 1 + r 1 + r 2 8 INTRODUCTION TO NETWORK ANALYSIS and 1 2 1 u − 1 2 + v − = 1.21b x x2
Equation (1.21a) represents a family of constant resistance circles. The centre of the circle for a normalised resistance r is at (r/ 1 + r 0) and the radius is 1/ 1+r . Equation (1.21b) describes a family of constant reactance circles. The centre of the circle with a normalised reactance x is at 1 1/x and the radius of the circle is 1/x. A simplified impedance Smith chart is shown in Figure 1.3. On the chart, the normalised resistance and reactance values, r and x, can be read when the reflection coefficient is plotted. On the other hand, the complex reflection coefficient can be determined when r and x values are known and plotted on the impedance chart.
1.1.3.2 Admittance chart
With the introduction of the normalised admittance Y y = = g + jb 1.22 Y0 ref where g and b are the normalised conductance and admittance, respectively. Equation (1.16) for the reflection coefficient can be rearranged to
g + jb − 1 − = 1.23 g + jb + 1
1
0.5 2
x = 0 r = ∞ r = 0 0.5 1 2
–0.5 –2 –1
Figure 1.3 Simplified Smith chart